Equirectangular skyboxes: Tips for avoiding pixellation?
4096 x 2048 should be adequate resolution, but keep in mind that the focal length of the camera you use in the second scene will make a big difference.

If your second scenes' camera focal length is high, 50-200mm, it is the same as opening up your equirectangular render in photoshop and zooming in a lot. Generally, a real HDR equirectangular image photographed in the real world will be 4k-12k in width. The 4K is best for reflections and lighting and the 12k might have some use as a seen sky in the 3D render, however, backplates, which are photographs taken at the same time at the same place using a different focal length (not a fisheye for panoramas, but more like a standard FOV (35mm-85mm) are often used to replace the environment image render. That image is then lined up to the rendercam (or should I say, the rendercam is aligned to the image) and it is composited in behind the CG elements, replacing the HDRI 'sky'.  If you wanted to do that without taking a separate photograph, your equirectangular environment map would have to be above 20k (generally speaking).

tl;dr  10-35mm blender camera settings should work fine with a 4k image if rendered out at 1080p.